Quotients of fake projective planes

TL;DR

This paper classifies the possible quotient surfaces and their minimal resolutions obtained from fake projective planes under various automorphism groups, extending understanding of their geometric structures.

Contribution

It provides a complete classification of quotient surfaces and their minimal resolutions for fake projective planes with specific automorphism groups.

Findings

Classified quotient surfaces for fake projective planes with automorphism groups Z/3Z, Z/7Z, 7:3, and (Z/3Z)^2.

Determined the structure of minimal resolutions of these quotient surfaces.

Extended the understanding of automorphism actions on fake projective planes.

Abstract

Recently, Prasad and Yeung classified all possible fundamental groups of fake projective planes. According to their result, many fake projective planes admit a nontrivial group of automorphisms, and in that case it is isomorphic to $\bbZ/3\bbZ$, $\bbZ/7\bbZ$, $7:3$, or $(\bbZ/3\bbZ)^2$, where $7:3$ is the unique non-abelian group of order 21. Let $G$ be a group of automorphisms of a fake projective plane $X$. In this paper we classify all possible structures of the quotient surface $X/G$ and its minimal resolution.